added 2 definitions

Author: JoramSoch

D/iass.md

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+---
+layout: definition
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2022-12-14 13:14:00
+
+title: "Interaction sum of squares"
+chapter: "Statistical Models"
+section: "Univariate normal data"
+topic: "Analysis of variance"
+definition: "Interaction sum of squares"
+
+sources:
+  - authors: "Nandy, Siddhartha"
+    year: 2018
+    title: "Two-Way Analysis of Variance"
+    in: "Stat 512: Applied Regression Analysis"
+    pages: "Purdue University, Summer 2018, Ch. 19"
+    url: "https://www.stat.purdue.edu/~snandy/stat512/topic7.pdf"
+
+def_id: "D184"
+shortcut: "iass"
+username: "JoramSoch"
+---
+
+
+**Definition:** Let there be an analysis of variance (ANOVA) model with [two](/D/anova2) or [more](/D/anovan) factors influencing the measured data $y$ (here, using the [standard formulation](/P/anova2-pss) of [two-way ANOVA](/D/anova2)):
+
+$$ \label{eq:anova}
+y_{ijk} = \mu + \alpha_i + \beta_j + \gamma_{ij} + \varepsilon_{ijk}, \; \varepsilon_{ijk} \overset{\mathrm{i.i.d.}}{\sim} \mathcal{N}(0, \sigma^2) \; .
+$$
+
+Then, the interaction sum of squares is defined as the [explained sum of squares] (ESS) for each interaction, i.e. as the sum of squared deviations of the average for each cell from the average across all observations, controlling for the [treatment sums of squares](/D/trss) of the corresponding factors:
+
+\begin{equation} \label{eq:iass}
+\begin{split}
+\mathrm{SS}_\mathrm{A \times B} &= \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} ([\bar{y}_{i j \bullet} - \bar{y}_{\bullet \bullet \bullet}] - [\bar{y}_{i \bullet \bullet} - \bar{y}_{\bullet \bullet \bullet}] - [\bar{y}_{\bullet j \bullet} - \bar{y}_{\bullet \bullet \bullet}])^2 \\
+&= \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} (\bar{y}_{i j \bullet} - \bar{y}_{i \bullet \bullet} - \bar{y}_{\bullet j \bullet} + \bar{y}_{\bullet \bullet \bullet})^2 \; .
+\end{split}
+\end{equation}
+
+Here, $\bar{y} _{i j \bullet}$ is the mean for the $(i,j)$-th cell (out of $a \times b$ cells), computed from $n_{ij}$ values $y_{ijk}$, $\bar{y} _{i \bullet \bullet}$ and $\bar{y} _{\bullet j \bullet}$ are the level means for the two factors and and $\bar{y} _{\bullet \bullet \bullet}$ is the mean across all values $y_{ijk}$.

D/trss.md

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+---
+layout: definition
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2022-12-14 13:01:00
+
+title: "Treatment sum of squares"
+chapter: "Statistical Models"
+section: "Univariate normal data"
+topic: "Analysis of variance"
+definition: "Treatment sum of squares"
+
+sources:
+  - authors: "Wikipedia"
+    year: 2022
+    title: "Analysis of variance"
+    in: "Wikipedia, the free encyclopedia"
+    pages: "retrieved on 2022-11-15"
+    url: "https://en.wikipedia.org/wiki/Analysis_of_variance#Partitioning_of_the_sum_of_squares"
+
+def_id: "D183"
+shortcut: "trss"
+username: "JoramSoch"
+---
+
+
+**Definition:** Let there be an analysis of variance (ANOVA) model with [one](/D/anova1), [two](/D/anova2) or [multiple](/D/anovan) factors influencing the measured data $y$ (here, using the [reparametrized version](/P/anova1-repara) of [one-way ANOVA](/D/anova1)):
+
+$$ \label{eq:anova}
+y_{ij} = \mu + \delta_i + \varepsilon_{ij}, \; \varepsilon_{ij} \overset{\mathrm{i.i.d.}}{\sim} \mathcal{N}(0, \sigma^2) \; .
+$$
+
+Then, the treatment sum of squares is defined as the [explained sum of squares] (ESS) for each main effect, i.e. as the sum of squared deviations of the average for each level of the factor, from the average across all observations:
+
+$$ \label{eq:trss}
+\mathrm{SS}_\mathrm{treat} = \sum_{i=1}^{k} \sum_{j=1}^{n_i} (\bar{y}_i - \bar{y})^2 \; .
+$$
+
+Here, $\bar{y}_i$ is the mean for the $i$-th level of the factor (out of $k$ levels), computed from $n_i$ values $y_{ij}$, and $\bar{y}$ is the mean across all values $y_{ij}$.